Is the Graph of a Continuous Map from X to Y a Closed Subset of XxY?

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In summary, the conversation discusses proving that the graph L={(x,y) in XxY: y=f(x)} is a closed subset of XxY, given that f:X->Y is a continuous map from a topological space X to a Hausdorff space Y. The participants mention that Hausdorff spaces are linked to open sets, and that a set is open if and only if its complement is closed. They suggest starting with L and considering different approaches to proving its closedness, and also mention that the assignment may not be due immediately. One participant asks for more hints, and another mentions a proof in a Banach space, but the others question its applicability.
  • #1
pivoxa15
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Homework Statement


Let f:X->Y be a cts map from a topological space X to a Hausdorff space Y. Prove that the graph L={(x,y) in XxY: y=f(x)} is a closed subset of XxY.

The Attempt at a Solution


Hausdorff space are linked with open sets so how do you prove closeness in XxY?
 
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  • #2
This is not important but since open sets are linked to closed sets, then anything to do with closed sets has some relation with open sets. You understand that a set is open if and only if its complement is closed?

So, start with L. What does it mean to be closed? There are many things to try, so write a few of them down and think about it for a while (perhaps a day or so, if need be - answers don't just magically appear instantly in people's minds).
 
  • #3
matt grime said:
There are many things to try, so write a few of them down and think about it for a while (perhaps a day or so, if need be - answers don't just magically appear instantly in people's minds).

If only the assignment is not due tomorrow morning.:rolleyes:
 
  • #4
L is closed if and only if its complement is open. Suppose that the complement is not open. What does that mean?
 
  • #5
matt grime said:
L is closed if and only if its complement is open. Suppose that the complement is not open. What does that mean?

Can you give more hint?

Why can't we use the proof in Banach space in the following link?
http://myyn.org/m/article/proof-of-closed-graph-theorem/
 
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1. What is a closed subset of XxY?

A closed subset of XxY is a subset of the Cartesian product of two sets, X and Y, that includes all of its limit points. This means that any sequence of points in the subset that converges in the product space XxY must also converge within the subset.

2. How is a closed subset different from an open subset?

A closed subset includes all of its limit points, whereas an open subset does not. This means that a closed subset contains all of the points that "surround" it, while an open subset does not include these boundary points.

3. Can a closed subset be empty?

Yes, a closed subset can be empty. This would occur if the subset does not contain any limit points, meaning that it is disjoint from the rest of the product space XxY.

4. What is the importance of closed subsets in mathematics?

Closed subsets are important in mathematics because they help define and distinguish different types of sets, such as open, closed, and compact sets. They also have applications in fields such as topology and analysis.

5. How can you determine if a subset is closed?

A subset is closed if it contains all of its limit points. This can be determined by checking if every convergent sequence in the subset also converges within the subset. Additionally, a subset is closed if its complement (the set of all points not in the subset) is open.

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